An alternating direction implicit method for orthogonal spline collocation linear systems

We consider the initial boundary value problem for a fractional evolution integro-differential equation with a weakly singular kernel arising from viscoelastic rods and membranes in a bounded smooth domain. In our numerical scheme, an arbitrary-order orthogonal spline collocation (OSC) method is used for spatial approximation and combined with the Crank-Nicolson (CN) alternating direction implicit (ADI) method for time approximation, and the integral term is approximated via the product integration (PI) rule. The detailed stability and convergence of the CN ADI OSC method in norm are proved. Finally, some numerical simulations are provided to illustrate our theoretical error estimates. In addition, the comparison between the non-uniform mesh and the uniform mesh shows the effectiveness of the non-uniform mesh when the solution is not sufficiently smooth.

The alternating direction implicit (ADI) method is a highly efficient technique for solving multi-dimensional dependent initial-boundary value problems on rectangles. Earlier we have used the ADI technique in conjunction with orthogonal spline collocation (OSC) for discretization in space to solve parabolic problems on rectangles and rectangular polygons. Recently, we extended applications of ADI OSC schemes to the solution of parabolic problems on some non-rectangular regions that allow for consistent nonuniform partitions. However, for many regions, it is impossible to construct such partitions. Therefore, in this paper, we show how to extend our approach further to solve parabolic problems on some non-rectangular regions using inconsistent uniform partitions. Numerical results are presented using piecewise Hermite cubic polynomials for spatial discretizations and our ADI OSC scheme for parabolic problems to demonstrate its performance on several regions.

The Crank–Nicolson orthogonal spline collocation method for one-dimensional parabolic problems with interfaces

ABSTRACTIn this paper, a second-order backward differentiation formula (BDF) orthogonal spline collocation (OSC) method with the truncation error of order in time and in space is presented for a kind of partial differential equation (PDE) with memory. The stability and convergence of the BDF OSC method in a new norm are proved. Besides, we provide three numerical experiments to demonstrate the results of theoretical analysis and show the accuracy and effectiveness of the BDF OSC method. Numerical experiments also exhibit the optimal error estimates in the , and-norms.

Anonoverlapping domain decomposition approach is used on uniform and matching grids to first define and then to compute the orthogonal spline collocation solution of the Dirichlet boundary value problem for Poisson's equation on an L-shaped region. We prove existence and uniqueness of the collocation solution and derive optimal order Hs-norm error bounds for s=0,1,2$. The collocation solution on two interfaces is computed using the preconditioned conjugate gradient method, and the collocation solution on three squares is computed by a matrix decomposition method that uses fast Fourier transforms. The total cost of the algorithm is O(N2 \log N), where the number of unknowns in the collocation solution is O(N2 )$.

A highly accurate algorithm for retrieving the predicted behavior of problems with piecewise-smooth initial data

We examine the use of orthogonal spline collocation for the semi-discreti\-za\-tion of the cubic Schr\"{o}dinger equation and the two-dimensional parabolic equation of Tappert. In each case, an optimal order\(L^2\) estimate of the error in the semidiscrete approximation is derived. For the cubic Schr\"{o}dinger equation, we present the results of numerical experiments in which the integration in time is performed using a routine from a software library.

We propose a multigrid Alternating-Direction Implicit (ADI) method for solving Maxwell's equations in this paper. This method is based on the interpretation of the ADI method as an iterative solver of the Crank-Nicolson (CN) scheme. By introducing a special solving procedure for the residual equation within the ADI framework, multigrid schemes are incorporated into the iterative ADI method. The accuracy and efficiency of the proposed multigrid ADI method are further demonstrated by numerical examples.

This paper describes a circuit-oriented leapfrog alternating direction implicit (ADI) method applied to a simulation of a power delivery network (PDN). An equivalent circuit model of the PDN is derived by using a recently-proposed conformal modeling technique, in which orthogonal meshes are used along with a treatment to improve the accuracy of the model. Since the conformal equivalent circuit model has a grid-like topology, it is suitable for direction-based algorithms such as an ADI method and the leapfrog ADI method. An advantage of the ADI-based method is that they can use a relatively-large time step size and reduce the total number of time steps in a transient simulation. In addition, the leapfrog ADI method is more efficient than the existing ADI method because its time marching procedure requires less updating processes. As a result, the conformal equivalent circuit model combined with the leapfrog ADI method can be accurate and efficient in modeling and analysis of the PDN.

The conventional three-dimensional alternating-direction implicit (ADI) method is modified by introducing an f factor (0 < f < 1). This modification allows the time step limit to be increased by a factor of 1/f with the solutions remaining stable and high accuracy being retained. This new method is tested for two different boundary conditions: a constant heat flux and a sudden heating of the surface to a constant temperature. In addition, it is compared with the popular Brian and Douglas methods, the results showing that the new ADI method has higher accuracy and requires less computer storage than those methods.

The approximate-factorisation-splitting (CNAFS) method as an efficient implementation of the Crank-Nicolson scheme for solving the three-dimensional Maxwell equations in the time domain, using much less CPU time and memory than a direct implementation, is presented. At each time step, the CNAFS method solves tridiagonal matrices successively instead of solving a huge sparse matrix. It is shown that CNAFS is unconditionally stable and has much smaller anisotropy than the alternating-direction implicit (ADI) method, though the numerical dispersion is the same as in the ADI method along the axes. In addition, for a given mesh density, there will be one value of the Courant number at which the CNAFS method has zero anisotropy, whereas the Crank-Nicolson scheme always has anisotropy. Analysis shows that both ADI and CNAFS have time step-size limits to avoid numerical attenuation, although both are still unconditionally stable beyond their limit.

Efficient second-order ADI difference schemes for three-dimensional Riesz space-fractional diffusion equations

ABSTRACTHeat transfer at microscale plays an important role in microtechnology applications. First, this paper is concerned with the numerical solution of a two-dimensional (2D) heat transfer equation at microscale through a second-order alternating direction implicit (ADI) method. By the discrete energy method, it is shown that the ADI solution converges to the exact solution with a convergence order of in L2-norm. Second, the ADI method is generalized to numerically solve the three-dimensional (3D) problem, and provides numerical solutions of order two in both time and space. Finally, numerical results testify the temporal and spatial accuracy of the ADI method.

ABSTRACTAs we know, second-order backward differentiation formula (BDF2) is L-stable scheme, which can damp unwanted finite oscillations, while Crank-Nicolson (C-N) method is only A-stable scheme and usually causes numerical oscillations. Thus, BDF2 may be more popular than C-N. However, up to now, most of studies focus on the error analysis of the compact alternating direction implicit (ADI) method on the basis of C-N method for temporal discretization in -norm. This paper is devoted to the convergence analysis of a compact multi-step ADI method used to solve a two-dimensional (2D) parabolic equation in -, - and -norms. Besides, the existence of the ADI solution is also analyzed in detail. A class of Richardson extrapolation algorithms are established to improve computational efficiency. By introducing a transformation, our algorithms are easily generalized to the solution of convection-diffusion problems with constant coefficients. Numerical results confirm the performance of our algorithms and support theoretical analysis.

A matched Peaceman–Rachford ADI method for solving parabolic interface problems